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The understanding of probability is complete when it relates not only with the concept’s definition, but also at the relationship between the mathematical model and the real world of random processes. The concept of  probability, even we refer strictly to the mathematical definition, is full of relativities and of philosophical and psychological implications. 

Relativity of probability

When we speak about the relativity of probability, we refer to the real objective way in which probability theory models the hazard and in which the human degree of belief in the occurrence of various events is sufficiently theoretically justified to make decisions. Thus, any criticism of the application of probability results in daily life will not hint at the mathematical theory itself, but at the transfer of theoretical information from the model to the surrounding reality. Briefly, these relativities are:

1)  Conceptual relativities
     a)  Terminological relativities
     – Mathematical probability and philosophical probability are different objects;
     b)  Relativities of mathematical definition
     – Defining a term through itself (the equally possible attribute from the classical definition);
     – The axiomatic nonstructural and nonindividual definition of event (as an element of a collective structure);
     – Choosing the set of axioms (Kolmogorov’s axiomatics in the complete definition);

2)  Relativities of equivalence of mathematical model with real world
     – The subject of philosophical probability is hazard and randomness, which cannot be mathematized;
     – Infinity, which is present in the definition of mathematical concept of probability, is not found again in the finite experimental reality;
     – The event, as a unit of mathematical theory, does not reproduce the event from the real world, which is much more complex;

3)  Relativities of practical applications of probability calculus
     – Choosing the field of events;
     – Idealizations of the equally probable type;
     – The subjective translation of the result of the Law of Large Numbers for finite successions of experiments.

     These relativities require at least an additional circumspection of the person who sees probability as an absolute degree of belief and implicitly the limitation of making decisions based on the numerical value of probability as a unique criterion.

Probability may be simultaneously viewed as:

 –  Limit of relative frequency within a sequence of tests performed under theoretically identical conditions;
      –  Objective measure of possibility;
      –  Subjective degree of belief in the occurrence of an event.

There are also other interpretations of probability, resulting from mathematical theories with similar structures:

 –  Predicted relative frequency within a physical model (Drieschner);
      –  Measure of tendency of an experimental context to produce an outcome (Popper);
      –  Logical relation between a data field and a hypothesis with respect to partial implication (Keynes);
      –  Numerical expression of an information about the existence of an event in certain conditions (Onicescu).

  All these interpretations are characterized by logical equivalences and contain elements having philosophical implications like prediction, possibility, frequency and degree of belief.

Philosophy of probability

What is the sense of the question: “What is the probability of …”? This seems to be the essential question around which all problems of philosophy of probability revolve. Great mathematicians like Pascal, Bernoulli, Laplace, Cornot, von Mises, Poincaré, Reichenbach, Popper, de Finetti, Carnap and Onicescu performed philosophical studies of the probability concept and dedicated to them an important part of their research, but the major questions still remain open to study:

     ●  Can probability also be defined in other terms besides through itself?

  Can we verify that it exists, at least in principle? What sense must be assigned to this existence? Does it express anything besides a lack of knowledge?

  Can a probability be assigned to an aleatory isolated event or just to some collective structures?

     These are just few of the basic questions that philosophy dealt with, through the efforts of the thinkers listed earlier, but still without a scientifically satisfactory conclusion. Hundreds of pages of papers might be written on each of such kind of questions.
     Probability has a double meaning: first as a measure of the real possibility of things (the physical probability revealed through frequency) and second as the degree of trust; in other words, there exist a philosophical probability and a mathematical one, and these are not to be confounded. The probability of an event does not really exist in the phenomenal world, like mass, force and the Greenwich meridian do not exist as real objects. It only exists abstractly. Its objective significance is that, starting from the same hypotheses, all mathematicians will find the same value for it, no matter the individual subjective opinions. It serves as a tool for acquiring a partial knowledge of the surrounding world, which is not equivalently and totally reproduced, simply because the hazard cannot be theoretically modeled and quantified. And then what is the justification for probability theory? What is the sense of its application? Humans are sentenced to act in uncertainty conditions. If humans had an infinite intelligence and calculus capacity, he or she could predict the future and would know our entire past. Probability theory is the mathematics of idealized hazard. Its application consists of reducing all events of a certain type to an arbitrary number of equally possible cases and calculating the number of favorable cases. Probability is nothing more than the mathematical degree of certainty we have about an event. It is simultaneously objective and subjective. Probability does not exist beyond us. In fact, it is not about the degree of certainty we have a priori, but one we should have if we were perfectly rational and could make the equally possible judgment. Therefore, probability is the only reasonable way to behave in conditions of partial knowledge and uncertainty, by using mathematics as a unique method, which is rigorous and unanimously accepted.



All above subjects are touched on in the book UNDERSTANDING AND CALCULATING THE ODDS: Probability Theory Basics and Calculus Guide for Beginners, with Applications in Games of Chance and Everyday Life. The dedicated chapters aim to stimulate the research and deep knowledge tendencies of readers with regard to these subjects, to complete an image of the probability concept that includes its philosophical and psychological aspects and to extend the simple image of a mathematical tool of calculus of degree of belief, which is so common among average people. You may find it in the Books section with a free sample.

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